Definition

The left adjoint is sometimes called the reflector, and a functor which is a reflector (or has a fully faithful right adjoint, which is the same up to equivalence) is called a reflection. Of course, there are dual notions of coreflective subcategory, coreflector, and coreflection.

Remark

A few sources (such as Categories Work) do not require a reflective subcategory to be full. However, in light of the fact that non-full subcategories are not invariant under equivalence, consideration of non-full reflective subcategories seems of limited usefulness. The general consensus among category theorists nowadays seems to be that “reflective subcategory” implies fullness. Examples for non-full subcategories and their behaviour can be found in a TAC paper by Adámek and Rosický.

Proof

This is a well-known set of equivalences concerning idempotent monads. The essential point is that a reflective subcategory i:B→Ai: B \to A is monadic, i.e., realizes BB as the category of algebras for the monadiri r on AA, where r:A→Br: A \to B is the reflector.

Complete reflective subcategories

When the unit of the reflector is a monomorphism, a reflective category is often thought of as a full subcategory of complete objects in some sense; the reflector takes each object in the ambient category to its completion. Such reflective subcategories are sometimes called mono-reflective. One similarly has epi-reflective (when the unit is an epimorphism) and bi-reflective (when the unit is a bimorphism).

Proposition

In this explicit form this appears as (Lurie, prop. 5.5.1.2). From (Adamek-Rosický) the “only if”-direction follows immediately from 2.53 there (saying that an accessibly embedded subcategory of an accessible category is accessible iff it is cone-reflective), while the “if”-direction follows immediately from 2.23 (saying any left or right adjoint between accessible categories is accessible).

Properties

General

A reflective subcategory is always closed under limits which exist in the ambient category (because the full inclusion is monadic, as noted above), and inherits colimits from the larger category by application of the reflector.

A morphism in a reflective subcategory is monic iff it is monic in the ambient category. A reflective subcategory of a well-powered category is well-powered.

Reflective subcategories of cartesian closed categories

In showing that a given category is cartesian closed, the following theorem is often useful (cf. A4.3.1 in the Elephant):

Theorem

If CC is cartesian closed, and D⊆CD\subseteq C is a reflective subcategory, then the reflector L:C→DL\colon C\to D preserves finite products if and only if DD is an exponential ideal (i.e. Y∈DY\in D implies YX∈DY^X\in D for any X∈CX\in C). In particular, if LL preserves finite products, then DD is cartesian closed.

Reflective and coreflective subcategories

Theorem

A subcategory of a category of presheaves[Aop,Set][A^{op}, Set] which is both reflective and coreflective is itself a category of presheaves [Bop,Set][B^{op}, Set], and the inclusion is induced by a functor A→BA \to B.

Property vs structure

Whenever CC is a full subcategory of DD, we can say that objects of CC are objects of DD with some extra property. But if CC is reflective in DD, then we can turn this around and (by thinking of the left adjoint as a forgetful functor) think of objects of DD as objects of CC with (if we're lucky) some extra structure or (in any case) some extra stuff.

This can always be made to work by brute force, but sometimes there is something insightful about it. For example, a metric space is a complete metric space equipped with a dense subset. Or, an integral domain is a field equipped with numerator and denominator functions.

Examples

Example

Complete metric spaces are mono-reflective in metric spaces; the reflector is called completion.

Example

Example

In a recollement situation, we have several reflectors and coreflectors. We have a reflective and coreflective subcategory i*:A′↪Ai_*: A' \hookrightarrow A with reflector i*i^* and coreflector i!i^!. The functor j*j^* is both a reflector for the reflective subcategory j*:A″↪Aj_*: A'' \hookrightarrow A, and a coreflector for the coreflective subcategory j!:A″↪Aj_!: A'' \hookrightarrow A.

Example

Assuming classical logic, the category Set has exactly three reflective (and replete) subcategories: the full subcategory containing all singleton sets; the full subcategory containing all subsingletons; and SetSet itself.

(Counter)Example

The non-full inclusion of unital rings into non-unital rings has a left adjoint (with monic units), whose reflector formally adjoins an identity element. However, we do not call it a reflective subcategory, because the “inclusion” is not full; see remark .

Remark

Notice that for R∈RingR \in Ring a ring with unit, its reflection LRL R in the above example is not in general isomorphic to RR, but is much larger. But an object in a reflective subcategory is necessarily isomorphic to its image under the reflector only if the reflective subcategory is full. While the inclusion Ring↪Ring\mathbf{Ring} \hookrightarrow \mathbf{Ring}‘ does have a left adjoint (as any forgetful functor between varieties of algebras, by the adjoint lifting theorem), this inclusion is not full (an arrow in Ring\mathbf{Ring}’ need not preserve the identity).